A034899 -id:A034899 - cp's OEIS frontend

A104500 Number of different groupings among the hierarchical orderings of n unlabeled elements.

1, 4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233, 1136383, 3116654, 8515706, 23172455, 62836916, 169801824, 457406173, 1228382159, 3289493887, 8784935160, 23400668297, 62179339101, 164832960183, 435978612329, 1150673925933, 3030701471118

Offset: 1

Thomas Wieder, Mar 11 2005

nonn

			Let * denote an element, let : denote separator among different levels within a hierarchy, let | denote a separator between different hierarchies. Furthermore, the braces {} indicate a group. For n=3 one has a(3) = 11 because
{***}, {*|*|*}, {*}{*}{*}, {*:*:*}, {*:**}, {*|**}, {*:*|*}, {*:*}{*}, {*|*}{*}, {**:*}, {*}{**}.

Alois P. Heinz, Table of n, a(n) for n = 1..800
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Cf. A034691, A104460, A034899.

etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=numtheory[divisors](j)) *b(n-j), j=1..n)/n) end end: b:= etr(n-> 2^(n-1)): a:= etr(b): seq(a(n), n=1..30); # Alois P. Heinz, Apr 21 2012

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b = etr[Function[{n}, 2^(n-1)]]; a = etr[b]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

Euler transform of 1, 3, 7, 18, 42, 104, 244, 585, 1373, ... = A034691.

More terms from Alois P. Heinz, Apr 21 2012

A158099 Euler transform of square powers of 2: [2,2^4,2^9,...,2^(n^2),...].

Original entry on oeis.org

1, 2, 19, 548, 66749, 33695574, 68787981855, 563088066184424, 18447871299903970005, 2417888543453357864445634, 1267655436282309648681395304255, 2658458526916981532120588021462151100, 22300750515466692968838881088968809185127601

Offset: 0

Paul D. Hanna, Mar 20 2009

nonn

			G.f.: A(x) = 1 + 2*x + 19*x^2 + 548*x^3 + 66749*x^4 +...
A(x) = 1/[(1-x)^2*(1-x^2)^(2^4)*(1-x^3)^(2^9)*(1-x^4)^(2^16)*...].

Alois P. Heinz, Table of n, a(n) for n = 0..35

Cf. A002416, A034899, A158098.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:= etr(n->2^(n^2)):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 03 2012

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Function[{n}, 2^(n^2)]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

a(n)=polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^(2^(k^2))),n)

{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d*2^(d^2))*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Oct 18 2009

G.f.: A(x) = 1/Product_{n>=1} (1 - x^n)^(2^(n^2)).

G.f.: exp( Sum_{n>=1} L(n)*x^n/n ) where L(n) = Sum_{d|n} d*2^(d^2). [Paul D. Hanna, Oct 18 2009]

A261043 Number of multisets of nonempty words with a total of n letters over binary alphabet such that all letters occur at least once in the multiset.

Original entry on oeis.org

0, 0, 3, 14, 49, 148, 427, 1170, 3150, 8288, 21562, 55368, 140998, 355854, 892014, 2220856, 5497483, 13533264, 33150801, 80825768, 196218139, 474423934, 1142756063, 2742781794, 6561049181, 15645058210, 37194447065, 88174246904, 208463588035, 491585765888

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Column k=2 of A257740.

Cf. A000041, A034899, A247003.

CoefficientList[Series[Product[1/(1-x^k)^(2^k), {k, 1, 30}] - 2*Product[1/(1 - x^k), {k, 1, 30}] + 1, {x, 0, 30}], x]
(* Second program: *)
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*k^# &]*A[n - j, k], {j, 1, n}]/n];
T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz in A257740 *)

a(n) = A034899(n) - 2*A000041(n) + 1.

a(n) ~ c^2 * 2^(n-1) * exp(2*sqrt(n) - 1/2) / (sqrt(Pi) * n^(3/4)), where c = A247003 = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502...

New name from Alois P. Heinz, Oct 07 2018

A104525 The number of hierarchical orderings among the parts of the integer partitions of the integer n.

Original entry on oeis.org

1, 4, 12, 40, 123, 395, 1227, 3851, 11944, 37032, 114144, 351040, 1075316, 3285398, 10007731, 30409157, 92169561, 278738219, 841132013, 2533138770, 7614144053, 22845435104, 68427663680, 204623945617, 610951554377, 1821438443615, 5422608839874, 16121857331124

Offset: 1

Thomas Wieder, Mar 12 2005. Definition revised Mar 28 2009

nonn

Euler transform of A055887 = number of ordered partitions of partitions.

			Let * denote an element, let : denote separator among different levels within a hierarchy, let | denote a separator between different hierarchies. Furthermore, the braces {} indicate a frame. For n=3 one has a(3) = 12 because:
{*:**}, {*:*}:{*}, {*}:{**}, {*:*:*}, {*}:{*}:{*}, {**}|{*}, {*}|{*:*}, {*}|{*}|{*}, {**}:{*}, {*}:{*:*}, {*}:{*}|{*}, {***}.

Alois P. Heinz, Table of n, a(n) for n = 1..750
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Thomas Wieder, Comments on A104525

Cf. A034691, A034899, A055887, A104460, A104500, A109186.

We can use combstruct to actually construct the structures A104525(n). %1 := Sequence(Set(Set(Z))).
with(combinat): with (numtheory): b:= proc(n) local k; option remember; `if`(n=0, 1, add (numbpart(k) * b(n-k), k=1..n)) end: a:= proc(n) option remember; `if` (n=0, 1, add (add (d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq (a(n), n=1..30); # Alois P. Heinz, Feb 02 2009

max = 30; A055887 = CoefficientList[1/(2 - 1/QPochhammer[x, x]) + O[x]^(max + 1), x] ; s = 1/Product[(1 - x^n)^A055887[[n + 1]], {n, 1, max}] + O[x]^max; CoefficientList[s, x] // Rest (* Jean-François Alcover, Jan 10 2016 *)

More terms from Alois P. Heinz, Feb 02 2009

A347011 Euler transform of j-> ceiling(2^(j-2)).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 93, 207, 453, 999, 2185, 4796, 10470, 22871, 49815, 108427, 235515, 511074, 1107248, 2396299, 5179169, 11181877, 24113939, 51949572, 111801422, 240381703, 516355235, 1108186951, 2376314763, 5091422730, 10900063776, 23317805916

Offset: 0

Alois P. Heinz, Aug 10 2021

nonn

Differs from A206301 first at n=10.

Alois P. Heinz, Table of n, a(n) for n = 0..3250

Cf. A034691, A034899, A166861, A187251, A206301, A319918.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, i-1)*binomial(ceil(2^(i-2))+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
       ceil(2^(d-2)), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);

CoefficientList[Series[1/(1-x) * Product[1/(1 - x^k)^(2^(k-2)), {k, 2, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 11 2021 *)

G.f.: Product_{j>0} 1/(1-x^j)^ceiling(2^(j-2)).

A104533 E.g.f.: exp(2x/(1-2x)).

Original entry on oeis.org

1, 2, 12, 104, 1168, 16032, 259264, 4817024, 100954368, 2353435136, 60355677184, 1687701792768, 51077784506368, 1662782678736896, 57917727119818752, 2148722382829027328, 84569896954751942656, 3518839711497761980416, 154306731918073225019392

Offset: 0

Thomas Wieder, Mar 13 2005

nonn

Number of hierarchical orderings for n labeled elements (see A075729) when there are two kinds A and B of elements.

			Let "a_i" and "b_j" be elements situated in the classes A and B with _i and _j as labels. Let : denote a separator among levels (ranks). Let | denote a separator among groups. E.g., a_1:b_2|b_1 is a hierarchy composed of two groups which contain three elements in total.
a(2) = 12 from b_2:b_1, b_2:a_1, b_2|b_1, a_1:a_2, b_2:a_1, a_1|a_2, a_1:b_2, a_2:a_1, b_1:a_2, a_2:b_1, b_1|a_2, b_2:b_1.

Seiichi Manyama, Table of n, a(n) for n = 0..398 (terms 0..150 from Alois P. Heinz)
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A075729, A034691, A034899.

Equals 2^n * A000262(n).

SetSeqUnnL := [T, {T=Set(S,card>=1), S=Sequence(U,card>=1), U=Union(a,b),a=Atom, b=Atom},labeled]; seq(count(SetSeqUnnL,size=j),j=1..20);
A104533 := proc(n::integer) local i,j,prttnlst,prttn,liste,ZahlVerschiedenerTeile,H,Mltplztt; Mltplztt:=vector[1000]; prttnlst:=partition(n); H := 0; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; liste := convert(prttn,multiset); ZahlVerschiedenerTeile := nops(liste); for j from 1 to ZahlVerschiedenerTeile do Mltplztt[j] := op(2,op(j,liste)); od; H := H + (n!/mul(Mltplztt[j]!,j=1..ZahlVerschiedenerTeile)) * 2^n; od; print(n,H); end proc;

CoefficientList[Exp[2 x/(1 - 2 x)] + O[x]^21, x]*Range[0, 20]!
(* or: *)
a[0] = 1; a[n_] := 2^n*n!*Hypergeometric1F1[n + 1, 2, 1]/E;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 10 2017 *)

a(n) = 2^n*A000262(n) = 2^n*n!*Sum_{k=0..n} C(n-1,k)/(k+1)!. - Paul Barry, Apr 28 2007
With p(n) = the number of integer partitions of n, d(i) = the number of different parts of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} n!/(prod_{j=1}^{d(i)} m(i, j)!) * 2^(n)
E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)*(1-2*x)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 09 2013
E.g.f.: E(0) - 1, where E(k) = 2 + 2*x/((2*k+1)*(1-2*x) - 2*x/E(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Dec 31 2013

Edited by N. J. A. Sloane, May 06 2008, at the suggestion of Joerg Arndt

A255969 E.g.f.: 1/Product_{k>=1} (1-x^k)^(x^k).

Original entry on oeis.org

1, 0, 2, 3, 44, 90, 2394, 6720, 202544, 1041768, 27369000, 170418600, 5835999432, 41711464080, 1489935696144, 14980499777880, 519279726915840, 5837621201012160, 232228922844775104, 2946339663605953920, 122979308145781345920, 1869847203939341074560

Offset: 0

Vaclav Kotesovec, Mar 12 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Vaclav Kotesovec, Asymptotics of sequence A034691

Cf. A034691, A034899.

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^(x^k),{k,1,nmax}],{x,0,nmax}],x] * Range[0, nmax]!
nmax=20; CoefficientList[Series[Exp[Sum[1/(k*(1/x^(k+1)-1)),{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!

log(a(n)) ~ 2*sqrt(n).

A256105 a(n) = [x^n] 2^(2*n) / Product_{k>=1} (1-x^k)^(2^(-k)).

Original entry on oeis.org

1, 2, 10, 36, 166, 556, 2724, 9000, 41542, 153164, 657644, 2325816, 11020508, 38006264, 164662664, 634362320, 2695771462, 9775537676, 43527018396, 156855914904, 687387270260, 2605392165928, 10799896586616, 40214700074800, 178809945153820, 657023566793400

Offset: 0

Vaclav Kotesovec, Mar 14 2015

nonn

Limit n->infinity a(n)^(1/n) = 4.

Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Vaclav Kotesovec, Graph a(n)/4^n

Cf. A034899, A144074.

Table[2^(2*n) * SeriesCoefficient[Product[1/(1-x^k)^(2^(-k)),{k,1,n}],{x,0,n}], {n,0,30}]
Table[4^n * (CoefficientList[Series[Exp[Sum[x^k/(2*k*(1-x^k/2)),{k,1,n}]],{x,0,n}],x])[[n+1]],{n,0,30}] (* faster *)

A334827 The number of oriented star-like and star trees with n arcs.

Original entry on oeis.org

4, 17, 66, 221, 688, 2034, 5788, 15998, 43192, 114496, 298712, 769340, 1959064, 4940761, 12354210, 30660947, 75583868, 185208833, 451356846, 1094522547, 2642121008, 6351335083, 15208854510, 36288478177, 86295204732, 204571273167, 483532711338, 1139738858221

Offset: 3

R. J. Mathar, Jun 09 2020

nonn

			a(6)=221 counts 132 oriented star-like trees with 3 rays and 6 arcs, 62 with 4 rays and 6 arcs, 20 with 5 rays and 6 arcs, and 7 star trees. In the illustrations in A000238 [Mathar] this is the same as 48 (shape 2) + 64 (shape 3) + 20 (shape 4) +32 (shape 7) + 30 (shape 8) +20 (shape 10) + 7 (shape 11).

Wikipedia, Starlike Tree
Index entries for sequences related to trees

Cf. A000238 (oriented trees), A051437 (linear oriented trees), A209406 (star-like oriented by number of arcs and rays), A004250 (undirected edges).

a(n) = A034899(n) -2^(n+1) = Sum_{k>=3} A209406(n,k).

cp's OEIS Frontend

A104500 Number of different groupings among the hierarchical orderings of n unlabeled elements.

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A158099 Euler transform of square powers of 2: [2,2^4,2^9,...,2^(n^2),...].

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A261043 Number of multisets of nonempty words with a total of n letters over binary alphabet such that all letters occur at least once in the multiset.

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A104525 The number of hierarchical orderings among the parts of the integer partitions of the integer n.

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A347011 Euler transform of j-> ceiling(2^(j-2)).

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A104533 E.g.f.: exp(2x/(1-2x)).

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A255969 E.g.f.: 1/Product_{k>=1} (1-x^k)^(x^k).

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